B. Maddah
ENMG 501 Engineering Management I
03/01/10
Probability and Random Variable (2)
Random Variables
Consider a function that assigns real numbers to events
(outcomes) in . Such real-valued function is a random
variable.
E.g., when rolling two fair

B. Maddah
ENMG 501 Class Project
04/20/11
CarBestBuy Inventory Management
You received the following letter from Layla Nassar
Dear OR consultant,
I am the CEO of CarBestBuythe leading car dealership in Hamra.
Recently we are facing a serious problem. Our

Dr. Maddah
ENMG 501 Review Quiz 1
02/20/07
OR stands for
a) Operating Room
b) Ossama Rahbani
c) Operations Research
d) Operational Research
A model is
a) A good fellow
b) A map or blueprint
c) An abstract representation of the real world
d) Cindy Crawfo

Dr. Maddah
ENMG 501 Review Quiz 2
02/20/07
The Sample space should be
a) Finite
b) Countable
c) Within (0, 1)
d) Within our solar system
Two events which are mutually exclusive are
a) A partition of
b) Independent
c) Such that both cant happen at the s

Dr. Maddah
ENMG 501 Review Quiz 3
06/03/07
If your friend has an exponential arrival time with
1/ = 1/20 mins then
a) He will mostly likely arrive after 30 mins
b) He is equally likely to arrive in any minute
c) Dont bother calling him when his late
d) D

Dr. Maddah
ENMG 501 Review Quiz 1
03/06/07
The number of flips until two heads appear is a
a) Geometric rv
b) Binomial rv
c) Positive Binomial rv
d) Negative Binomial rv
The following rvs has the memoryless property
a) The binomial
b) The geometric
c) T

Dr. Maddah
ENMG 501 Review Quiz 5
20/03/07
Which of the following is a MC?
a) Xn where Pcfw_Xn = 1| Xn1 = 1, Xn2 = 0 = Pcfw_Xn = 1
b) Xn where Pcfw_Xn = 1| Xn1 = 1, Xn2 = 0 = Pcfw_Xn = 1|Xn2 = 0
c) Xn where Pcfw_Xn = 1| Xn1 = 1, Xn2 = 0 = Pcfw_Xn = 1|Xn1

Dr. Maddah
ENMG 501 Review Quiz 6
20/03/07
An absorbing state is
a) A state where a MC gets trapped for ever
b) Not transient
c) California
d) Recurrent
If all states communicate in a MC with N < states, then
a) The MC is irreducible
b) The MC has one c

Dr. Maddah
ENMG 501 Review Quiz 7
03/05/07
In all Markovian queues the underlying CTMC chain is
a) The number of people in queue at time t, Lq(t).
b) The number of people in the system at time t, L(t).
c) The time in queue, Wq(t).
d) The time in system,

Dr. Maddah
ENMG 501 Review Quiz 8
03/05/07
In M/M/c, the mean number of busy servers is
a) = /(c).
b) a = /.
c) L Lq .
d) < c.
A M/M/c with arrival rate and service rate is
a) The same as a M/M/1 with same and service rate c
b) Better than a M/M/1 with

B. Maddah
ENMG 622 ENMG 501
05/27/07
Inventory Theory (2)
The single-period newsvendor model
Consider a newsvendor, who, at the start of each day, must
decide the amount of newspapers to stock, S.
Placing an order has a negligible cost.
Daily demand for

B. Maddah
ENMG 622 ENMG 501
05/27/07
Inventory Theory (1)
Introduction
Most industries have to deal with inventories. E.g., shelf
and warehouse (back room) inventory in retail, and raw
material, work in process and finished product inventory in
manufactu

S1. Suppose trials, each having a probability p of being a success, are performed until k
success occur. What is the probability that n trials are performed?
S2. On any given day, there are n customers in the market. Out of these customers, a
customer ent

S1. Let E, F, G be three events. Find expressions for the events that out of E, F, G,
(a) only F occurs,
(b) both E and F but not G occurs,
(c) at least one event occurs,
(d) at least two events occur,
(e) all three events occur,
(f) none occurs,
(g) at m

B. Maddah
ENMG 501 Engineering Management I
02/22/10
Probability and Random Variable (1)
Sample space and Events
Suppose that an experiment with an uncertain outcome is
performed (e.g., rolling a die).
While the outcome of the experiment is not known in

B. Maddah
ENMG 501 Engineering Management I
03/06/07
Discrete Time Markov Chains (1)
Stochastic processes
A stochastic (random) process, Xt, t T, is a collection
(family) of random variables, where T is an index set.
A stochastic process is a representat

B. Maddah
ENMG 501 Engineering Management I
02/27/07
Probability and Random Variable (3)
The Geometric Random Variable
Suppose independent trials, each having a probability p of
being a success, are performed.
We define the geometric random variable (rv)

B. Maddah
ENMG 501 Engineering Management I
02/27/07
4
Probability and Random Variable (5)
Lognormal random variable A stock price model
A rv Y is said to be lognormal if X = ln(Y) is a normal
random variable.
Alternatively, Y is a lognormal rv if Y = eX

B. Maddah
ENMG 501 Engineering Management I
02/16/09
The OR Modeling Approach
What is special about the OR analysis approach?
(i) A primary focus on decision making. The analysis must lead
to clear suggestions to the decision maker.
(ii) An appraisal res

B. Maddah
ENMG 501 Engineering Management I
03/17/07
Discrete Time Markov Chains (3)
Long-run Properties of MC (Stationary Solution)
Consider the two-state MC of the weather condition in
Example 4.
0.5749 0.4251
0.5715 0.4285
P4 =
P8 =
0.5668 0.4

B. Maddah
ENMG 501 Engineering Management I
03/12/07
Discrete Time Markov Chains (2)
Chapman-Kolmogorov (C-K) Equations
(n)
Consider a MC, Xn with state space cfw_0,1, 2, . Let pij be
the n-step transition probability from state i to state j. I.e.,
(
pij

B. Maddah
ENMG 622 ENMG 501
03/27/07
Queueing Theory (1)
What is a queueing system?
A queueing system consists of servers (resources) that
provide service to customers (entities).
A Customer requesting service will start service if the
required server is

B. Maddah
ENMG 501 Engineering Management I
03/17/07
Continuous-Time Markov Chains
Definition and Connection to the Exponential Distribution
A continuous-time stochastic process cfw_X(t), t 0 taking on
positive integers is said to be a continuous-time Ma

B. Maddah
ENMG 622 ENMG 501
05/20/09
Queueing Theory (2)
Distribution of waiting time in M/M/1
Let Tq be the waiting time in queue of a customer.
Then it can be shown that,
Pcfw_Tq > t = e ( )t .
Let T be the total time of a customer in the system (in qu

B. Maddah
ENMG 622 ENMG 501
05/17/10
Queueing Theory (3)
The M/M/c/K queue
This is a generalization of M/M/1/K to many servers.
Specifically, this is a Markovian queue with c servers and
K c waiting spaces (where K > c).
The number of customers in the M/

Markov Fun: Rebecca
On any particular day Rebecca is either cheerful or gloomy. If she
is cheerful today then she will be cheerful tomorrow with
probability 0.7. If she is gloomy today then she will be gloomy
tomorrow with probability 0.4.
(a) If Rebecca

American University of Beirut
Faculty of Engineering and Architecture
Engineering Management Program
ENMG 501 Engineering Management II
Spring 2013, CRN 20078: TTh 11:00 AM - 12:15 PM, Bechtel 206
Instructor
Dr. Bacel Maddah
Oce: Bechtel 519A
E-mail: bace